Fast Least SquaresFast Least Squares MigrationMigrationwith a Deblurring Filterwith a Deblurring Filter

30 October 2008

Naoshi Aoki

1

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical result of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

2

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical result of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

3

Forward and Inverse ProblemsForward and Inverse Problemsfor Acoustic Wavefieldfor Acoustic Wavefield

• Forward problem:

where d is data, L is forward modeling operator, and m is reflectivity model.

• Inverse problem:

where LT is an adjoint of forward modeling operator, and [LTL]-1 is the inverse of Hessian.

,d Lm

, -1T Tm L L L d

4

Alternatives to Direct InversionAlternatives to Direct Inversion

• Migration

• LSM (e.g., Nemeth, Wu and Schuster,1999)

where

Tmigm = L d

1 ,n n n m =m g

,n Tng L (Lm -d)

,

,n n

nn n

g g

Lg Lg

T= L Lm

5

The The UU Model Test Model Test

3D U Model Model Description• Model size:

– 1.8 x 1.8 x 1.8 km

• U shape reflectivity anomaly

• Cross-spread geometry– Source : 16 shots, 100 m int.– Receiver : 16 receivers , 100 m int.

Depth (m) Reflectivity

250 1

500 -1

750 1

1000 -1

1250 1

● Source● Receiver

U model is designed for testing Prestack 3D LSM with arbitrary 3D survey geometry.

Data0

5

TW

T (

s)

0 1.8X (m)

6

Depth Slices fromDepth Slices fromMigration and LSMMigration and LSM

(c) Z = 250 m (e) Z = 750 m (g) Z=1250m(a) Actual Reflectivity

Kirchhoff Migration Images

(b) Test geometry(d) Z=250m

LSM Images after 30 Iterations(f) Z=750m (h) Z=1250m

● Source● Receiver

7

Challenges in LSM ProcessingChallenges in LSM Processing

• Estimation of modeling operators– Velocity Model– Source wavelet

• Computational Cost– LSM typically requires 10 or more iterations.– Each LSM iteration requires about 3 times

higher computational cost than that of the migration.

8

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical result of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

9

An Alternative to LSMAn Alternative to LSM

• Deblur the migration image with a local non-stationary filtering– Migration deconvolution (Hu and Schuster,

2001),– Deconvolution of migration operator by a local

non-stationary filter (Etgen, 2002, Guitton 2004),

– FFT based approach(e.g., Lecomte(2008); Toxopeus et al, (2008)).

10

Deblurring Filter TheoryDeblurring Filter Theory• Actual Migration Image:

• Compute a reference migration image from a reference model m’:

• Find a deblurring operator with a matching filter (He, 2003) :

• Apply the operator to the actual migration image

T TL d = L Lm

' Td'F L =m

'T TL d' = L Lm

TdF L m

-1TLF L

The computational cost is about one iteration of LSM

11

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical result of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

12

0

2.5

Z (

km)

0 2.5X (km)

0.1-0.1 0

Actual Reflectivity Model

Point Scatterer Model TestPoint Scatterer Model Test

TW

T (

sec)

X (km)0.5 1.5

1.8

2.8

CSG Example

Fdominant = 5 Hz; λ=200 m

Scatterer:50 m x 50 m

V=1000 m/s

▼▼▼▼▼▼▼▼▼▼▼▼▼

13

Migration ImageMigration Image

0

2.5

Z (

km)

0 2.5X (km)

Actual Reflectivity Image

Z (

km)

0

2.50 2.5

X (km)

Migration Image

0.1-0.1 0The Rayleigh resolution limit = 200 m 14

Deblurred Migration ImageDeblurred Migration Image

0

2.5

Z (

km)

0 2.5X (km)

Actual Reflectivity Image

0

2.5

Z (

km)

0 2.5X (km)

Deblurred Migration Image

0.1-0.1 015

LSM ImageLSM Image

0

2.5

Z (

km)

0 2.5X (km)

Actual Reflectivity Image

0.1-0.1 0

0

2.5

Z (

km)

0 2.5X (km)

LSM Image after 30 Iterations

16

Horizontal Image of the ScattererHorizontal Image of the Scatterer

0.1

0

0.5 1.5

Ref

lect

ivity

X(km)17

Migration Deblurring Test Summary Migration Deblurring Test Summary

• Deblurring filter improves spatial resolution of migration image about double.

• The computational cost is about one iteration of LSM.

• The deblurred migration image is slightly noisier than that in the LSM image.

18

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical results of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

19

Deblurred LSM TheoryDeblurred LSM Theory

• DLSM is a fast LSM with a deblurring filter.• 2 types of DLSM algorithms are proposed:

1. Regularized DLSM (or RDLSM)

where mapri is a skeletonized version of ,

and γ is a regularization parameter.

2. Preconditioned DLSM (or PDLSM)

1 ,n n n m =m g

,n Tn aprig L (Lm -d) mm -

TFL d

1 ,n n nm =m Fg 2

,.n n

n

n

g g

gFL

F

2

2 2 ,n

n

n

g

Lg g

20

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical results of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

21

Numerical ResultsNumerical Results

• A synthetic data set from the Marmousi2 model.

• A 2D marine data set from the Gulf of Mexico.

22

Marmousi2 ModelMarmousi2 ModelGeological Cross SectionGeological Cross Section

(Martin et. al., 2006)(Martin et. al., 2006) 23

Velocity and Density ModelsVelocity and Density Models

0

3

0 15

Z (

km)

X (km)

P wave Velocity Model

4.51.5Velocity (km/s)

0

3

0 15Z

(km

)

X (km)

Density Model

2.61Density (g/cc) 24

Traveltime Field ComputationTraveltime Field Computation

0

3

0 15

Z (

km)

X (km)

P wave Velocity Model

4.51.5Velocity (km/s)

0

3

0 15Z

(km

)

X (km)

Traveltime Field Example

41 Velocity (km/s)

(UTAM ray- tracing code written by He, 2002)25

Reflectivity Model and DataReflectivity Model and Data

0 300Time (msec)

0

2000

-2000A

mpl

itude

Source WaveletReflectivity Model

0

3

Z (

km)

6 12X (km)

0.2-0.2 0

Fdom = 25 Hz

26

Reflectivity Model and DataReflectivity Model and Data

Zero-offset Data

0

3T

WT

(s)

6 12X (km)

Reflectivity Model

0

3

Z (

km)

6 12X (km)

0.2-0.2 0 27

Migration ImageMigration Image

Poststack Migration

0

3Z

(km

)

6 12X (km)

Actual Reflectivity Model

0

3

Z (

km)

6 12X (km)

0.2-0.2 0

CPU time = 10 minutes

on a dual processor 2.2 GHz

Velocity: 1800-4500 m/sWavelength : 70 - 180 m

28

Deblurring Filter with the Exact Model Deblurring Filter with the Exact Model Step1: Compute Matching OperatorStep1: Compute Matching Operator

Actual Migration Image

0

3Z

(km

)

6 12X (km)

Exact Model

0

3

Z (

km)

6 12X (km)

f

29

Deblurring Filter with the Exact Model Deblurring Filter with the Exact Model Step2: Apply the OperatorStep2: Apply the Operator

Deblurred Migration Image

0

3Z

(km

)

6 12X (km)

Actual Migration Image

0

3

Z (

km)

6 12X (km)

f

30

DLSM Convergence CurvesDLSM Convergence Curves

PDLSMPDLSM1

01 30

Iteration Number

Res

idua

l

1

01 30

Iteration Number

Res

idua

l

819

Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30

RDLSMRDLSM

31

DLSM ImagesDLSM Images with the Exact Model with the Exact Model

0

3Z

(km

)

6 12X (km)

PDLSM after 8 Iterations0

3

Z (

km)

6 12X (km)

RDLSM after 19 Iterations

32

Model Sensitivity TestModel Sensitivity Test• Exact model:

– the actual model

• Geological model:– Skeletonized Migrated

Image

• Grid model:– The region is divided into

sections; each section has a point scatterer in the center.

Exact Model

0

3Z

(km

)

6 12X (km)

Geological Model

0

3Z

(km

)

6 12X (km)

Zoom View of Grid Model

1

2Z

(km

)

10 11X (km)

250 x 250 m

33

Deblurring Filter with the Geological Model Deblurring Filter with the Geological Model Step1: Compute Matching OperatorStep1: Compute Matching Operator

Reference Migration Image

0

3Z

(km

)

6 12X (km)

Geological Model

0

3

Z (

km)

6 12X (km)

f

34

Deblurring Filter with the Geological Model Deblurring Filter with the Geological Model Step2: Apply the OperatorStep2: Apply the Operator

Deblurred Migration Image

0

3Z

(km

)

6 12X (km)

Actual Migration Image

0

3

Z (

km)

6 12X (km)

f

35

DLSM Convergence CurvesDLSM Convergence Curves

Preconditioned DLSMPreconditioned DLSM

1

01 30

Iteration Number

Res

idua

l

Regularized DLSMRegularized DLSM

1

01 30

Iteration Number

Res

idua

l

20 12

Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30 36

DLSM ImagesDLSM Images with the Geological Model with the Geological Model

0

3Z

(km

)

6 12X (km)

PDLSM after 12 Iterations0

3

Z (

km)

6 12X (km)

RDLSM after 20 Iterations

37

Zoom View of Grid Model

1

2

Z (

km)

10 11X (km)

Deblurring Filter with the Grid Model Deblurring Filter with the Grid Model Step1: Compute Matching OperatorStep1: Compute Matching Operator

Reference Migration Image0

3Z

(km

)

6 12X (km)

f

38

Deblurring Filter with the Grid Model Deblurring Filter with the Grid Model Step2: Apply the OperatorStep2: Apply the Operator

Deblurred Migration Image

0

3Z

(km

)

6 12X (km)

Actual Migration Image

0

3

Z (

km)

6 12X (km)

f

39

Regularized DLSMRegularized DLSM

1

01 30

Iteration Number

Res

idua

l

Damping parameter: Γ= 200000x0.5n-1, n=1,2,…,30

DLSM Convergence CurvesDLSM Convergence Curves

Preconditioned DLSMPreconditioned DLSM

1

01 30

Iteration Number

Res

idua

l20 10

40

DLSM ImagesDLSM Images with the Grid Model with the Grid Model

0

3

Z (

km)

6 12X (km)

RDLSM after 20 Iterations0

3Z

(km

)

6 12X (km)

PDLSM after 10 Iterations

41

Marmousi2 Test Summary (1)Marmousi2 Test Summary (1)

• The deblurring filter can expedite the computation of an LSM image.– RDLSM and PDLSM provide acceptable LSM images

with about 2/3 and 1/3 the cost of standard LSM, respectively.

• Controlling the model dependency is required.– RDLSM can control the model dependency with a

regularization parameter.

– In the PDLSM algorithm, not using a deblurring filter after several iteration is recommended.

42

Marmousi2 Test Summary (2)Marmousi2 Test Summary (2)

• DLSM with the geological model– Computation of an LSM image can be expedited by a

human interpretation.– A risk is an erroneous interpretation. The model

dependency should be carefully controlled.

• DLSM with the grid model– The result is not good as that from a better geological

model. – An advantage is that no expense of a human interpretation

is required for the model building.

43

The Gulf of Mexico Data TestThe Gulf of Mexico Data Test

84

TW

T(s

)

X (km)18

02D Poststack Marine Data

44

The Gulf of Mexico Data TestThe Gulf of Mexico Data Test

• Both the regularization and preconditioning schemes are employed in the DLSM.

• A geological model is created by the following way:1.A deblurred migration image is obtained with a grid

model.

2.A geological model is created by cosmetic filtering and skeletonizing the deblurred migration image.

45

Zero-offset Data from Zero-offset Data from for a Grid Modelfor a Grid Model

84

TW

T(ｓ

)

X (km)18

0

Scatterer Interval: 500 m x 500 m

46

Zoom View of Reference Migration Zoom View of Reference Migration Image for a Grid ModelImage for a Grid Model

8

1.2

Z (

km)

X (km)

1310.5

0.4

47

Kirchhoff MigrationKirchhoff Migration

8

1

1.5

Z (

km)

X (km)

1310.5

0.5

48

Deblurred Migration ImageDeblurred Migration ImageZ

(km

)

X (km)

8

1

1.51310.5

0.5

49

Geological ModelGeological Model

8

1

1.5

Z (

km)

X (km)

1310.5

0.5

0

0.1

-0.1

Reflectivity

50

Comparison of Imaging ResultsComparison of Imaging Results

0.5

1.5

Z (

km)

8 13X (km)

Kirchhoff Migration

51

Box A: Comparison of ImagesBox A: Comparison of Images

0.5

0.7

Z (

km)

9.6 10.6X (km)

Migration

0.5

0.7

Z (

km)

9.6 10.6X (km)

LSM after 3 Iterations

0.5

0.7

Z (

km)

9.6 10.6X (km)

DLSM after 3 Iterations

0.5

0.7

Z (

km)

9.6 10.6X (km)

LSM after 10 Iterations

52

Box B: Comparison of ImagesBox B: Comparison of ImagesMigration

1

1.2

Z (

km)

11 12X (km)

LSM after 3 Iterations

1.2

Z (

km)

11 12X (km)

1

DLSM after 3 Iterations

1.2Z (

km)

11 12X (km)

1

LSM after 10 Iterations

1.2

Z (

km)

11 12X (km)

1

53

Total Computational CostTotal Computational CostMigration

1

1.2

Z (

km)

11 12X (km)

LSM after 3 Iterations

1.2

Z (

km)

11 12X (km)

1

DLSM after 3 Iterations

1.2Z (

km)

11 12X (km)

1

LSM after 10 Iterations

1.2

Z (

km)

11 12X (km)

1

1 9

19+ 3054

Total Computational CostTotal Computational Cost

• Migration 1• LSM 3 Iterations 9• LSM 10 Iterations 30• DLSM 3 Iterations 19+

– Deblurring with the grid model 3– Deblurring with the geological model 4+– DLSM 3 Iterations 12

55

The GOM Data Test SummaryThe GOM Data Test Summary

• DLSM can successfully provide an improved LSM image with an affordable computer expense.

56

OutlinesOutlines

• Motivation

• Deblurring filter theory

• A numerical results of the deblurring filter

• Deblurred LSM theory

• Numerical results of the deblurred LSM

• Conclusions

57

ConclusionsConclusions

• A deblurring filter provides a fine apriori model for a regularized LSM, and it can also be used as an effective preconditioning filter.

• The DLSM algorithms provids acceptable LSM images with 1/3 – 2/3 the cost of standard LSM.

58

Future WorksFuture Works

• The deblurring filter requires some improvement in quality and efficiency.

• A computer-aided skeletonization method is required for reducing an expense of a human interpretation.

59

AcknowledgementsAcknowledgements• I would like to thank Prof. Gerard T. Schuster for

his encouragement throughout my stay at the University of Utah.

• I also want to thank my group colleagues for their academic discussions and personal help.

• I also thank JOGMEC and JAPEX for supporting my study at the University of Utah.

60

ThanksThanks

61